Runtime Complexity of the query pattern
preorder(g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Clauses:

preorder(X, Y) :- pdl(X, -(Y, [])).
pdl(nil, Y) :- ','(!, eq(Y, -(X, X))).
pdl(T, -(.(X, Xs), Zs)) :- ','(value(T, X), ','(left(T, L), ','(right(T, R), ','(pdl(L, -(Xs, Ys)), pdl(R, -(Ys, Zs)))))).
left(nil, nil).
left(tree(L, X1, X2), L).
right(nil, nil).
right(tree(X3, X4, R), R).
value(nil, X5).
value(tree(X6, X, X7), X).
eq(X, X).

Query: preorder(g,a)

(1) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: preorder(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: preorder(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: pdl(T9, -(T11, []))\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: pdl(T9, -(T11, [])), SCOPE: 2, CLAUSE: 1 | pdl(T9, -(T11, [])), SCOPE: 2, CLAUSE: 2\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_2, eq(-(T16, []), -(X22, X22))) | pdl(nil, -(T11, [])), SCOPE: 2, CLAUSE: 2\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: pdl(T9, -(T11, [])), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nX21 is free\npdl(T9, -(T11, [])) !=? pdl(nil, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: eq(-(T16, []), -(X22, X22))\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: eq(-(T16, []), -(X22, X22)), SCOPE: 3, CLAUSE: 9\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(value(T26, T29), ','(left(T26, X41), ','(right(T26, X42), ','(pdl(X41, -(T30, X43)), pdl(X42, -(X43, []))))))\n\nT26 is ground\nX21 is free\nX41 is free\nX42 is free\nX43 is free\npdl(T26, -(T11, [])) !=? pdl(nil, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(value(T26, T29), ','(left(T26, X41), ','(right(T26, X42), ','(pdl(X41, -(T30, X43)), pdl(X42, -(X43, [])))))), SCOPE: 4, CLAUSE: 7 | ','(value(T26, T29), ','(left(T26, X41), ','(right(T26, X42), ','(pdl(X41, -(T30, X43)), pdl(X42, -(X43, [])))))), SCOPE: 4, CLAUSE: 8\n\nT26 is ground\nX21 is free\nX41 is free\nX42 is free\nX43 is free\npdl(T26, -(T11, [])) !=? pdl(nil, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(value(T26, T29), ','(left(T26, X41), ','(right(T26, X42), ','(pdl(X41, -(T30, X43)), pdl(X42, -(X43, [])))))), SCOPE: 4, CLAUSE: 8\n\nT26 is ground\nX21 is free\nX41 is free\nX42 is free\nX43 is free\npdl(T26, -(T11, [])) !=? pdl(nil, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(left(tree(T39, T40, T41), X41), ','(right(tree(T39, T40, T41), X42), ','(pdl(X41, -(T42, X43)), pdl(X42, -(X43, [])))))\n\nT39 is ground\nT40 is ground\nT41 is ground\nX41 is free\nX42 is free\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(left(tree(T39, T40, T41), X41), ','(right(tree(T39, T40, T41), X42), ','(pdl(X41, -(T42, X43)), pdl(X42, -(X43, []))))), SCOPE: 5, CLAUSE: 3 | ','(left(tree(T39, T40, T41), X41), ','(right(tree(T39, T40, T41), X42), ','(pdl(X41, -(T42, X43)), pdl(X42, -(X43, []))))), SCOPE: 5, CLAUSE: 4\n\nT39 is ground\nT40 is ground\nT41 is ground\nX41 is free\nX42 is free\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(left(tree(T39, T40, T41), X41), ','(right(tree(T39, T40, T41), X42), ','(pdl(X41, -(T42, X43)), pdl(X42, -(X43, []))))), SCOPE: 5, CLAUSE: 4\n\nT39 is ground\nT40 is ground\nT41 is ground\nX41 is free\nX42 is free\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(right(tree(T51, T52, T53), X42), ','(pdl(T51, -(T42, X43)), pdl(X42, -(X43, []))))\n\nT51 is ground\nT52 is ground\nT53 is ground\nX42 is free\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(right(tree(T51, T52, T53), X42), ','(pdl(T51, -(T42, X43)), pdl(X42, -(X43, [])))), SCOPE: 6, CLAUSE: 5 | ','(right(tree(T51, T52, T53), X42), ','(pdl(T51, -(T42, X43)), pdl(X42, -(X43, [])))), SCOPE: 6, CLAUSE: 6\n\nT51 is ground\nT52 is ground\nT53 is ground\nX42 is free\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(right(tree(T51, T52, T53), X42), ','(pdl(T51, -(T42, X43)), pdl(X42, -(X43, [])))), SCOPE: 6, CLAUSE: 6\n\nT51 is ground\nT52 is ground\nT53 is ground\nX42 is free\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(pdl(T62, -(T42, X43)), pdl(T64, -(X43, [])))\n\nT62 is ground\nT64 is ground\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: pdl(T62, -(T42, X43))\n\nT62 is ground\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: pdl(T64, -(T66, []))\n\nT64 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: pdl(T62, -(T42, X43)), SCOPE: 7, CLAUSE: 1 | pdl(T62, -(T42, X43)), SCOPE: 7, CLAUSE: 2\n\nT62 is ground\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(!_7, eq(-(T71, X100), -(X99, X99))) | pdl(nil, -(T42, X43)), SCOPE: 7, CLAUSE: 2\n\nX43 is free\nX100 is free\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: pdl(T62, -(T42, X43)), SCOPE: 7, CLAUSE: 2\n\nT62 is ground\nX43 is free\nX98 is free\npdl(T62, -(T42, X43)) !=? pdl(nil, X98)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: eq(-(T71, X100), -(X99, X99))\n\nX100 is free\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: eq(-(T71, X100), -(X99, X99)), SCOPE: 8, CLAUSE: 9\n\nX100 is free\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(value(T81, T84), ','(left(T81, X124), ','(right(T81, X125), ','(pdl(X124, -(T85, X126)), pdl(X125, -(X126, X127))))))\n\nT81 is ground\nX43 is free\nX98 is free\nX127 is free\nX124 is free\nX125 is free\nX126 is free\npdl(T81, -(T42, X43)) !=? pdl(nil, X98)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(value(T81, T84), ','(left(T81, X124), ','(right(T81, X125), ','(pdl(X124, -(T85, X126)), pdl(X125, -(X126, X127)))))), SCOPE: 9, CLAUSE: 7 | ','(value(T81, T84), ','(left(T81, X124), ','(right(T81, X125), ','(pdl(X124, -(T85, X126)), pdl(X125, -(X126, X127)))))), SCOPE: 9, CLAUSE: 8\n\nT81 is ground\nX43 is free\nX98 is free\nX127 is free\nX124 is free\nX125 is free\nX126 is free\npdl(T81, -(T42, X43)) !=? pdl(nil, X98)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(value(T81, T84), ','(left(T81, X124), ','(right(T81, X125), ','(pdl(X124, -(T85, X126)), pdl(X125, -(X126, X127)))))), SCOPE: 9, CLAUSE: 8\n\nT81 is ground\nX43 is free\nX98 is free\nX127 is free\nX124 is free\nX125 is free\nX126 is free\npdl(T81, -(T42, X43)) !=? pdl(nil, X98)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(left(tree(T94, T95, T96), X124), ','(right(tree(T94, T95, T96), X125), ','(pdl(X124, -(T97, X126)), pdl(X125, -(X126, X127)))))\n\nT94 is ground\nT95 is ground\nT96 is ground\nX127 is free\nX124 is free\nX125 is free\nX126 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(left(tree(T94, T95, T96), X124), ','(right(tree(T94, T95, T96), X125), ','(pdl(X124, -(T97, X126)), pdl(X125, -(X126, X127))))), SCOPE: 10, CLAUSE: 3 | ','(left(tree(T94, T95, T96), X124), ','(right(tree(T94, T95, T96), X125), ','(pdl(X124, -(T97, X126)), pdl(X125, -(X126, X127))))), SCOPE: 10, CLAUSE: 4\n\nT94 is ground\nT95 is ground\nT96 is ground\nX127 is free\nX124 is free\nX125 is free\nX126 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(left(tree(T94, T95, T96), X124), ','(right(tree(T94, T95, T96), X125), ','(pdl(X124, -(T97, X126)), pdl(X125, -(X126, X127))))), SCOPE: 10, CLAUSE: 4\n\nT94 is ground\nT95 is ground\nT96 is ground\nX127 is free\nX124 is free\nX125 is free\nX126 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(right(tree(T106, T107, T108), X125), ','(pdl(T106, -(T97, X126)), pdl(X125, -(X126, X127))))\n\nT106 is ground\nT107 is ground\nT108 is ground\nX127 is free\nX125 is free\nX126 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(right(tree(T106, T107, T108), X125), ','(pdl(T106, -(T97, X126)), pdl(X125, -(X126, X127)))), SCOPE: 11, CLAUSE: 5 | ','(right(tree(T106, T107, T108), X125), ','(pdl(T106, -(T97, X126)), pdl(X125, -(X126, X127)))), SCOPE: 11, CLAUSE: 6\n\nT106 is ground\nT107 is ground\nT108 is ground\nX127 is free\nX125 is free\nX126 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(right(tree(T106, T107, T108), X125), ','(pdl(T106, -(T97, X126)), pdl(X125, -(X126, X127)))), SCOPE: 11, CLAUSE: 6\n\nT106 is ground\nT107 is ground\nT108 is ground\nX127 is free\nX125 is free\nX126 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(pdl(T117, -(T97, X126)), pdl(T119, -(X126, X127)))\n\nT117 is ground\nT119 is ground\nX127 is free\nX126 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: pdl(T117, -(T97, X126))\n\nT117 is ground\nX126 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: pdl(T119, -(T121, X127))\n\nT119 is ground\nX127 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 79 [arrowhead = none ];
79 -> 3;

80 [label="ONLY EVAL with clause\npreorder(X14, X15) :- pdl(X14, -(X15, [])).\nand substitutionT1 -> T9,\nX14 -> T9,\nT2 -> T11,\nX15 -> T11,\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 80 [arrowhead = none ];
80 -> 7;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 81 [arrowhead = none ];
81 -> 10;

82 [label="EVAL with clause\npdl(nil, X21) :- ','(!_2, eq(X21, -(X22, X22))).\nand substitutionT9 -> nil,\nT11 -> T16,\nX21 -> -(T16, []),\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 82 [arrowhead = none ];
82 -> 13;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 83 [arrowhead = none ];
83 -> 14;

84 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 84 [arrowhead = none ];
84 -> 15;

85 [label="EVAL with clause\npdl(X37, -(.(X38, X39), X40)) :- ','(value(X37, X38), ','(left(X37, X41), ','(right(X37, X42), ','(pdl(X41, -(X39, X43)), pdl(X42, -(X43, X40)))))).\nand substitutionT9 -> T26,\nX37 -> T26,\nX38 -> T29,\nX39 -> T30,\nT11 -> .(T29, T30),\nX40 -> [],\nT27 -> T29,\nT28 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 85 [arrowhead = none ];
85 -> 24;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 86 [arrowhead = none ];
86 -> 26;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 87 [arrowhead = none ];
87 -> 16;

88 [label="EVAL with clause\neq(X25, X25).\nand substitutionT16 -> [],\nX25 -> -([], []),\nX22 -> [],\nT19 -> []", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 88 [arrowhead = none ];
88 -> 20;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 89 [arrowhead = none ];
89 -> 21;

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 90 [arrowhead = none ];
90 -> 22;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 91 [arrowhead = none ];
91 -> 28;

92 [label="BACKTRACK\nfor clause: value(nil, X5)\nwith clash: (pdl(T26, -(T11, [])), pdl(nil, X21))", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 92 [arrowhead = none ];
92 -> 30;

93 [label="EVAL with clause\nvalue(tree(X54, X55, X56), X55).\nand substitutionX54 -> T39,\nX55 -> T40,\nX56 -> T41,\nT26 -> tree(T39, T40, T41),\nT29 -> T40,\nT30 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 93 [arrowhead = none ];
93 -> 31;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 94 [arrowhead = none ];
94 -> 32;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 95 [arrowhead = none ];
95 -> 33;

96 [label="BACKTRACK\nfor clause: left(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 96 [arrowhead = none ];
96 -> 34;

97 [label="ONLY EVAL with clause\nleft(tree(X69, X70, X71), X69).\nand substitutionT39 -> T51,\nX69 -> T51,\nT40 -> T52,\nX70 -> T52,\nT41 -> T53,\nX71 -> T53,\nX41 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 97 [arrowhead = none ];
97 -> 39;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 98 [arrowhead = none ];
98 -> 40;

99 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 99 [arrowhead = none ];
99 -> 41;

100 [label="ONLY EVAL with clause\nright(tree(X84, X85, X86), X86).\nand substitutionT51 -> T62,\nX84 -> T62,\nT52 -> T63,\nX85 -> T63,\nT53 -> T64,\nX86 -> T64,\nX42 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 100 [arrowhead = none ];
100 -> 42;

101 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
42 -> 101 [arrowhead = none ];
101 -> 43;

102 [label="SPLIT 2\nnew knowledge:\nT62 is ground\nreplacements:X43 -> T66", fontsize=14, style = filled, fillcolor = violet];
42 -> 102 [arrowhead = none ];
102 -> 44;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 103 [arrowhead = none ];
103 -> 46;

104 [label="INSTANCE with matching:\nT9 -> T64\nT11 -> T66", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 104 [arrowhead = none , style=dashed];
104 -> 7[style=dashed];

105 [label="EVAL with clause\npdl(nil, X98) :- ','(!_7, eq(X98, -(X99, X99))).\nand substitutionT62 -> nil,\nT42 -> T71,\nX43 -> X100,\nX98 -> -(T71, X100),\nT70 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 105 [arrowhead = none ];
105 -> 47;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 106 [arrowhead = none ];
106 -> 48;

107 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 107 [arrowhead = none ];
107 -> 49;

108 [label="EVAL with clause\npdl(X120, -(.(X121, X122), X123)) :- ','(value(X120, X121), ','(left(X120, X124), ','(right(X120, X125), ','(pdl(X124, -(X122, X126)), pdl(X125, -(X126, X123)))))).\nand substitutionT62 -> T81,\nX120 -> T81,\nX121 -> T84,\nX122 -> T85,\nT42 -> .(T84, T85),\nX43 -> X127,\nX123 -> X127,\nT82 -> T84,\nT83 -> T85", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 108 [arrowhead = none ];
108 -> 55;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 109 [arrowhead = none ];
109 -> 56;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 110 [arrowhead = none ];
110 -> 50;

111 [label="ONLY EVAL with clause\neq(X105, X105).\nand substitutionT71 -> T74,\nX100 -> T74,\nX105 -> -(T74, T74),\nX99 -> T74,\nX106 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 111 [arrowhead = none ];
111 -> 51;

112 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 112 [arrowhead = none ];
112 -> 52;

113 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 113 [arrowhead = none ];
113 -> 58;

114 [label="BACKTRACK\nfor clause: value(nil, X5)\nwith clash: (pdl(T81, -(T42, X43)), pdl(nil, X98))", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 114 [arrowhead = none ];
114 -> 59;

115 [label="EVAL with clause\nvalue(tree(X138, X139, X140), X139).\nand substitutionX138 -> T94,\nX139 -> T95,\nX140 -> T96,\nT81 -> tree(T94, T95, T96),\nT84 -> T95,\nT85 -> T97", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 115 [arrowhead = none ];
115 -> 62;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 116 [arrowhead = none ];
116 -> 63;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 117 [arrowhead = none ];
117 -> 64;

118 [label="BACKTRACK\nfor clause: left(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 118 [arrowhead = none ];
118 -> 65;

119 [label="ONLY EVAL with clause\nleft(tree(X153, X154, X155), X153).\nand substitutionT94 -> T106,\nX153 -> T106,\nT95 -> T107,\nX154 -> T107,\nT96 -> T108,\nX155 -> T108,\nX124 -> T106", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 119 [arrowhead = none ];
119 -> 66;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 120 [arrowhead = none ];
120 -> 67;

121 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 121 [arrowhead = none ];
121 -> 68;

122 [label="ONLY EVAL with clause\nright(tree(X168, X169, X170), X170).\nand substitutionT106 -> T117,\nX168 -> T117,\nT107 -> T118,\nX169 -> T118,\nT108 -> T119,\nX170 -> T119,\nX125 -> T119", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 122 [arrowhead = none ];
122 -> 74;

123 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
74 -> 123 [arrowhead = none ];
123 -> 77;

124 [label="SPLIT 2\nnew knowledge:\nT117 is ground\nreplacements:X126 -> T121", fontsize=14, style = filled, fillcolor = violet];
74 -> 124 [arrowhead = none ];
124 -> 78;

125 [label="INSTANCE with matching:\nT62 -> T117\nT42 -> T97\nX43 -> X126", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 125 [arrowhead = none , style=dashed];
125 -> 43[style=dashed];

126 [label="INSTANCE with matching:\nT62 -> T119\nT42 -> T121\nX43 -> X127", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 126 [arrowhead = none , style=dashed];
126 -> 43[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f1_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f42_in(z0, z2), tree(z0, z1, z2))
U2(f42_out1, tree(z0, z1, z2)) → f7_out1
f43_in(nil) → f43_out1
f43_in(tree(z0, z1, z2)) → U3(f74_in(z0, z2), tree(z0, z1, z2))
U3(f74_out1, tree(z0, z1, z2)) → f43_out1
f42_in(z0, z1) → U4(f43_in(z0), z0, z1)
U4(f43_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f42_out1
f74_in(z0, z1) → U6(f43_in(z0), z0, z1)
U6(f43_out1, z0, z1) → U7(f43_in(z1), z0, z1)
U7(f43_out1, z0, z1) → f74_out1

Tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(U2'(f42_in(z0, z2), tree(z0, z1, z2)), F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(U3'(f74_in(z0, z2), tree(z0, z1, z2)), F74_IN(z0, z2))
F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
U4'(f43_out1, z0, z1) → c9(U5'(f7_in(z1), z0, z1), F7_IN(z1))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
U6'(f43_out1, z0, z1) → c12(U7'(f43_in(z1), z0, z1), F43_IN(z1))

S tuples:

F1_IN(z0) → c(U1'(f7_in(z0), z0), F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(U2'(f42_in(z0, z2), tree(z0, z1, z2)), F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(U3'(f74_in(z0, z2), tree(z0, z1, z2)), F74_IN(z0, z2))
F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
U4'(f43_out1, z0, z1) → c9(U5'(f7_in(z1), z0, z1), F7_IN(z1))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
U6'(f43_out1, z0, z1) → c12(U7'(f43_in(z1), z0, z1), F43_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, f43_in, U3, f42_in, U4, U5, f74_in, U6, U7

Defined Pair Symbols:

F1_IN, F7_IN, F43_IN, F42_IN, U4', F74_IN, U6'

Compound Symbols:

c, c3, c6, c8, c9, c11, c12

(3) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f1_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f42_in(z0, z2), tree(z0, z1, z2))
U2(f42_out1, tree(z0, z1, z2)) → f7_out1
f43_in(nil) → f43_out1
f43_in(tree(z0, z1, z2)) → U3(f74_in(z0, z2), tree(z0, z1, z2))
U3(f74_out1, tree(z0, z1, z2)) → f43_out1
f42_in(z0, z1) → U4(f43_in(z0), z0, z1)
U4(f43_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f42_out1
f74_in(z0, z1) → U6(f43_in(z0), z0, z1)
U6(f43_out1, z0, z1) → U7(f43_in(z1), z0, z1)
U7(f43_out1, z0, z1) → f74_out1

Tuples:

F7_IN(tree(z0, z1, z2)) → c3(U2'(f42_in(z0, z2), tree(z0, z1, z2)), F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(U3'(f74_in(z0, z2), tree(z0, z1, z2)), F74_IN(z0, z2))
F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
U4'(f43_out1, z0, z1) → c9(U5'(f7_in(z1), z0, z1), F7_IN(z1))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
U6'(f43_out1, z0, z1) → c12(U7'(f43_in(z1), z0, z1), F43_IN(z1))
F1_IN(z0) → c1(U1'(f7_in(z0), z0))
F1_IN(z0) → c1(F7_IN(z0))

S tuples:

F7_IN(tree(z0, z1, z2)) → c3(U2'(f42_in(z0, z2), tree(z0, z1, z2)), F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(U3'(f74_in(z0, z2), tree(z0, z1, z2)), F74_IN(z0, z2))
F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
U4'(f43_out1, z0, z1) → c9(U5'(f7_in(z1), z0, z1), F7_IN(z1))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
U6'(f43_out1, z0, z1) → c12(U7'(f43_in(z1), z0, z1), F43_IN(z1))
F1_IN(z0) → c1(U1'(f7_in(z0), z0))
F1_IN(z0) → c1(F7_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, f43_in, U3, f42_in, U4, U5, f74_in, U6, U7

Defined Pair Symbols:

F7_IN, F43_IN, F42_IN, U4', F74_IN, U6', F1_IN

Compound Symbols:

c3, c6, c8, c9, c11, c12, c1

(5) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f1_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f42_in(z0, z2), tree(z0, z1, z2))
U2(f42_out1, tree(z0, z1, z2)) → f7_out1
f43_in(nil) → f43_out1
f43_in(tree(z0, z1, z2)) → U3(f74_in(z0, z2), tree(z0, z1, z2))
U3(f74_out1, tree(z0, z1, z2)) → f43_out1
f42_in(z0, z1) → U4(f43_in(z0), z0, z1)
U4(f43_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f42_out1
f74_in(z0, z1) → U6(f43_in(z0), z0, z1)
U6(f43_out1, z0, z1) → U7(f43_in(z1), z0, z1)
U7(f43_out1, z0, z1) → f74_out1

Tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F1_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))
F1_IN(z0) → c1

S tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F1_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))
F1_IN(z0) → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, f43_in, U3, f42_in, U4, U5, f74_in, U6, U7

Defined Pair Symbols:

F42_IN, F74_IN, F1_IN, F7_IN, F43_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(7) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN(z0) → c1(F7_IN(z0))
F1_IN(z0) → c1

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f1_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f42_in(z0, z2), tree(z0, z1, z2))
U2(f42_out1, tree(z0, z1, z2)) → f7_out1
f43_in(nil) → f43_out1
f43_in(tree(z0, z1, z2)) → U3(f74_in(z0, z2), tree(z0, z1, z2))
U3(f74_out1, tree(z0, z1, z2)) → f43_out1
f42_in(z0, z1) → U4(f43_in(z0), z0, z1)
U4(f43_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f42_out1
f74_in(z0, z1) → U6(f43_in(z0), z0, z1)
U6(f43_out1, z0, z1) → U7(f43_in(z1), z0, z1)
U7(f43_out1, z0, z1) → f74_out1

Tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F1_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))
F1_IN(z0) → c1

S tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))

K tuples:

F1_IN(z0) → c1(F7_IN(z0))
F1_IN(z0) → c1

Defined Rule Symbols:

f1_in, U1, f7_in, U2, f43_in, U3, f42_in, U4, U5, f74_in, U6, U7

Defined Pair Symbols:

F42_IN, F74_IN, F1_IN, F7_IN, F43_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))

We considered the (Usable) Rules:

f43_in(nil) → f43_out1
f43_in(tree(z0, z1, z2)) → U3(f74_in(z0, z2), tree(z0, z1, z2))
f74_in(z0, z1) → U6(f43_in(z0), z0, z1)
U3(f74_out1, tree(z0, z1, z2)) → f43_out1
U6(f43_out1, z0, z1) → U7(f43_in(z1), z0, z1)
U7(f43_out1, z0, z1) → f74_out1

And the Tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F1_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))
F1_IN(z0) → c1

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F1_IN(x₁)) = [1] + [2]x₁   
POL(F42_IN(x₁, x₂)) = [2]x₂   
POL(F43_IN(x₁)) = 0   
POL(F74_IN(x₁, x₂)) = 0   
POL(F7_IN(x₁)) = [2]x₁   
POL(U3(x₁, x₂)) = 0   
POL(U4'(x₁, x₂, x₃)) = [2]x₃   
POL(U6(x₁, x₂, x₃)) = [2]x₃   
POL(U6'(x₁, x₂, x₃)) = 0   
POL(U7(x₁, x₂, x₃)) = x₃   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f43_in(x₁)) = 0   
POL(f43_out1) = 0   
POL(f74_in(x₁, x₂)) = [2] + [2]x₂   
POL(f74_out1) = 0   
POL(nil) = 0   
POL(tree(x₁, x₂, x₃)) = [2] + x₃   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f1_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f42_in(z0, z2), tree(z0, z1, z2))
U2(f42_out1, tree(z0, z1, z2)) → f7_out1
f43_in(nil) → f43_out1
f43_in(tree(z0, z1, z2)) → U3(f74_in(z0, z2), tree(z0, z1, z2))
U3(f74_out1, tree(z0, z1, z2)) → f43_out1
f42_in(z0, z1) → U4(f43_in(z0), z0, z1)
U4(f43_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f42_out1
f74_in(z0, z1) → U6(f43_in(z0), z0, z1)
U6(f43_out1, z0, z1) → U7(f43_in(z1), z0, z1)
U7(f43_out1, z0, z1) → f74_out1

Tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F1_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))
F1_IN(z0) → c1

S tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))

K tuples:

F1_IN(z0) → c1(F7_IN(z0))
F1_IN(z0) → c1
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))

Defined Rule Symbols:

f1_in, U1, f7_in, U2, f43_in, U3, f42_in, U4, U5, f74_in, U6, U7

Defined Pair Symbols:

F42_IN, F74_IN, F1_IN, F7_IN, F43_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(11) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f1_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f42_in(z0, z2), tree(z0, z1, z2))
U2(f42_out1, tree(z0, z1, z2)) → f7_out1
f43_in(nil) → f43_out1
f43_in(tree(z0, z1, z2)) → U3(f74_in(z0, z2), tree(z0, z1, z2))
U3(f74_out1, tree(z0, z1, z2)) → f43_out1
f42_in(z0, z1) → U4(f43_in(z0), z0, z1)
U4(f43_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f42_out1
f74_in(z0, z1) → U6(f43_in(z0), z0, z1)
U6(f43_out1, z0, z1) → U7(f43_in(z1), z0, z1)
U7(f43_out1, z0, z1) → f74_out1

Tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F1_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))
F1_IN(z0) → c1

S tuples:

F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))

K tuples:

F1_IN(z0) → c1(F7_IN(z0))
F1_IN(z0) → c1
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))

Defined Rule Symbols:

f1_in, U1, f7_in, U2, f43_in, U3, f42_in, U4, U5, f74_in, U6, U7

Defined Pair Symbols:

F42_IN, F74_IN, F1_IN, F7_IN, F43_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))

We considered the (Usable) Rules:

f43_in(nil) → f43_out1
f43_in(tree(z0, z1, z2)) → U3(f74_in(z0, z2), tree(z0, z1, z2))
f74_in(z0, z1) → U6(f43_in(z0), z0, z1)
U3(f74_out1, tree(z0, z1, z2)) → f43_out1
U6(f43_out1, z0, z1) → U7(f43_in(z1), z0, z1)
U7(f43_out1, z0, z1) → f74_out1

And the Tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F1_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))
F1_IN(z0) → c1

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F1_IN(x₁)) = [3]x₁   
POL(F42_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(F43_IN(x₁)) = x₁   
POL(F74_IN(x₁, x₂)) = x₁ + x₂   
POL(F7_IN(x₁)) = [3]x₁   
POL(U3(x₁, x₂)) = 0   
POL(U4'(x₁, x₂, x₃)) = [3]x₃   
POL(U6(x₁, x₂, x₃)) = [2] + [2]x₂   
POL(U6'(x₁, x₂, x₃)) = x₃   
POL(U7(x₁, x₂, x₃)) = [2] + x₂   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f43_in(x₁)) = 0   
POL(f43_out1) = 0   
POL(f74_in(x₁, x₂)) = [2] + [2]x₁   
POL(f74_out1) = [2]   
POL(nil) = 0   
POL(tree(x₁, x₂, x₃)) = [1] + x₁ + x₃   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f7_in(z0), z0)
U1(f7_out1, z0) → f1_out1
f7_in(nil) → f7_out1
f7_in(tree(z0, z1, z2)) → U2(f42_in(z0, z2), tree(z0, z1, z2))
U2(f42_out1, tree(z0, z1, z2)) → f7_out1
f43_in(nil) → f43_out1
f43_in(tree(z0, z1, z2)) → U3(f74_in(z0, z2), tree(z0, z1, z2))
U3(f74_out1, tree(z0, z1, z2)) → f43_out1
f42_in(z0, z1) → U4(f43_in(z0), z0, z1)
U4(f43_out1, z0, z1) → U5(f7_in(z1), z0, z1)
U5(f7_out1, z0, z1) → f42_out1
f74_in(z0, z1) → U6(f43_in(z0), z0, z1)
U6(f43_out1, z0, z1) → U7(f43_in(z1), z0, z1)
U7(f43_out1, z0, z1) → f74_out1

Tuples:

F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
F1_IN(z0) → c1(F7_IN(z0))
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))
F1_IN(z0) → c1

S tuples:

F74_IN(z0, z1) → c11(U6'(f43_in(z0), z0, z1), F43_IN(z0))
U6'(f43_out1, z0, z1) → c12(F43_IN(z1))

K tuples:

F1_IN(z0) → c1(F7_IN(z0))
F1_IN(z0) → c1
F7_IN(tree(z0, z1, z2)) → c3(F42_IN(z0, z2))
F42_IN(z0, z1) → c8(U4'(f43_in(z0), z0, z1), F43_IN(z0))
U4'(f43_out1, z0, z1) → c9(F7_IN(z1))
F43_IN(tree(z0, z1, z2)) → c6(F74_IN(z0, z2))

Defined Rule Symbols:

f1_in, U1, f7_in, U2, f43_in, U3, f42_in, U4, U5, f74_in, U6, U7

Defined Pair Symbols:

F42_IN, F74_IN, F1_IN, F7_IN, F43_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(15) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: preorder(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: preorder(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: pdl(T5, -(T7, []))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: pdl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 1 | pdl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(!_2, eq(-(T11, []), -(X16, X16))) | pdl(nil, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: pdl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nX15 is free\npdl(T5, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: eq(-(T11, []), -(X16, X16))\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(-(T11, []), -(X16, X16)), SCOPE: 3, CLAUSE: 9\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, []))))))\n\nT21 is ground\nX15 is free\nX35 is free\nX36 is free\nX37 is free\npdl(T21, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, [])))))), SCOPE: 4, CLAUSE: 7 | ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, [])))))), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX15 is free\nX35 is free\nX36 is free\nX37 is free\npdl(T21, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, [])))))), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX15 is free\nX35 is free\nX36 is free\nX37 is free\npdl(T21, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, [])))))\n\nT34 is ground\nT35 is ground\nT36 is ground\nX35 is free\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, []))))), SCOPE: 5, CLAUSE: 3 | ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, []))))), SCOPE: 5, CLAUSE: 4\n\nT34 is ground\nT35 is ground\nT36 is ground\nX35 is free\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, []))))), SCOPE: 5, CLAUSE: 4\n\nT34 is ground\nT35 is ground\nT36 is ground\nX35 is free\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, []))))\n\nT46 is ground\nT47 is ground\nT48 is ground\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, [])))), SCOPE: 6, CLAUSE: 5 | ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, [])))), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT47 is ground\nT48 is ground\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, [])))), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT47 is ground\nT48 is ground\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(pdl(T57, -(T37, X37)), pdl(T59, -(X37, [])))\n\nT57 is ground\nT59 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: pdl(T57, -(T37, X37))\n\nT57 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: pdl(T59, -(T61, []))\n\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: pdl(T57, -(T37, X37)), SCOPE: 7, CLAUSE: 1 | pdl(T57, -(T37, X37)), SCOPE: 7, CLAUSE: 2\n\nT57 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(!_7, eq(-(T66, X94), -(X93, X93))) | pdl(nil, -(T37, X37)), SCOPE: 7, CLAUSE: 2\n\nX37 is free\nX94 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: pdl(T57, -(T37, X37)), SCOPE: 7, CLAUSE: 2\n\nT57 is ground\nX37 is free\nX92 is free\npdl(T57, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: eq(-(T66, X94), -(X93, X93))\n\nX94 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: eq(-(T66, X94), -(X93, X93)), SCOPE: 8, CLAUSE: 9\n\nX94 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121))))))\n\nT76 is ground\nX37 is free\nX92 is free\nX121 is free\nX118 is free\nX119 is free\nX120 is free\npdl(T76, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121)))))), SCOPE: 9, CLAUSE: 7 | ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121)))))), SCOPE: 9, CLAUSE: 8\n\nT76 is ground\nX37 is free\nX92 is free\nX121 is free\nX118 is free\nX119 is free\nX120 is free\npdl(T76, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121)))))), SCOPE: 9, CLAUSE: 8\n\nT76 is ground\nX37 is free\nX92 is free\nX121 is free\nX118 is free\nX119 is free\nX120 is free\npdl(T76, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121)))))\n\nT89 is ground\nT90 is ground\nT91 is ground\nX121 is free\nX118 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121))))), SCOPE: 10, CLAUSE: 3 | ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121))))), SCOPE: 10, CLAUSE: 4\n\nT89 is ground\nT90 is ground\nT91 is ground\nX121 is free\nX118 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121))))), SCOPE: 10, CLAUSE: 4\n\nT89 is ground\nT90 is ground\nT91 is ground\nX121 is free\nX118 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121))))\n\nT101 is ground\nT102 is ground\nT103 is ground\nX121 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121)))), SCOPE: 11, CLAUSE: 5 | ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121)))), SCOPE: 11, CLAUSE: 6\n\nT101 is ground\nT102 is ground\nT103 is ground\nX121 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121)))), SCOPE: 11, CLAUSE: 6\n\nT101 is ground\nT102 is ground\nT103 is ground\nX121 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(pdl(T112, -(T92, X120)), pdl(T114, -(X120, X121)))\n\nT112 is ground\nT114 is ground\nX121 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: pdl(T112, -(T92, X120))\n\nT112 is ground\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: pdl(T114, -(T116, X121))\n\nT114 is ground\nX121 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 93 [arrowhead = none ];
93 -> 4;

94 [label="ONLY EVAL with clause\npreorder(X10, X11) :- pdl(X10, -(X11, [])).\nand substitutionT1 -> T5,\nX10 -> T5,\nT2 -> T7,\nX11 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 94 [arrowhead = none ];
94 -> 5;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 95 [arrowhead = none ];
95 -> 6;

96 [label="EVAL with clause\npdl(nil, X15) :- ','(!_2, eq(X15, -(X16, X16))).\nand substitutionT5 -> nil,\nT7 -> T11,\nX15 -> -(T11, []),\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 96 [arrowhead = none ];
96 -> 8;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 97 [arrowhead = none ];
97 -> 9;

98 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 98 [arrowhead = none ];
98 -> 11;

99 [label="EVAL with clause\npdl(X31, -(.(X32, X33), X34)) :- ','(value(X31, X32), ','(left(X31, X35), ','(right(X31, X36), ','(pdl(X35, -(X33, X37)), pdl(X36, -(X37, X34)))))).\nand substitutionT5 -> T21,\nX31 -> T21,\nX32 -> T24,\nX33 -> T25,\nT7 -> .(T24, T25),\nX34 -> [],\nT22 -> T24,\nT23 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 99 [arrowhead = none ];
99 -> 23;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 100 [arrowhead = none ];
100 -> 25;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 101 [arrowhead = none ];
101 -> 12;

102 [label="EVAL with clause\neq(X19, X19).\nand substitutionT11 -> [],\nX19 -> -([], []),\nX16 -> [],\nT14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 102 [arrowhead = none ];
102 -> 17;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 103 [arrowhead = none ];
103 -> 18;

104 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 104 [arrowhead = none ];
104 -> 19;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 105 [arrowhead = none ];
105 -> 27;

106 [label="BACKTRACK\nfor clause: value(nil, X5)\nwith clash: (pdl(T21, -(T7, [])), pdl(nil, X15))", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 106 [arrowhead = none ];
106 -> 29;

107 [label="EVAL with clause\nvalue(tree(X48, X49, X50), X49).\nand substitutionX48 -> T34,\nX49 -> T35,\nX50 -> T36,\nT21 -> tree(T34, T35, T36),\nT24 -> T35,\nT25 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 107 [arrowhead = none ];
107 -> 35;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 108 [arrowhead = none ];
108 -> 36;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 109 [arrowhead = none ];
109 -> 37;

110 [label="BACKTRACK\nfor clause: left(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 110 [arrowhead = none ];
110 -> 38;

111 [label="ONLY EVAL with clause\nleft(tree(X63, X64, X65), X63).\nand substitutionT34 -> T46,\nX63 -> T46,\nT35 -> T47,\nX64 -> T47,\nT36 -> T48,\nX65 -> T48,\nX35 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 111 [arrowhead = none ];
111 -> 45;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 112 [arrowhead = none ];
112 -> 53;

113 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 113 [arrowhead = none ];
113 -> 54;

114 [label="ONLY EVAL with clause\nright(tree(X78, X79, X80), X80).\nand substitutionT46 -> T57,\nX78 -> T57,\nT47 -> T58,\nX79 -> T58,\nT48 -> T59,\nX80 -> T59,\nX36 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 114 [arrowhead = none ];
114 -> 57;

115 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
57 -> 115 [arrowhead = none ];
115 -> 60;

116 [label="SPLIT 2\nnew knowledge:\nT57 is ground\nreplacements:X37 -> T61", fontsize=14, style = filled, fillcolor = violet];
57 -> 116 [arrowhead = none ];
116 -> 61;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 117 [arrowhead = none ];
117 -> 69;

118 [label="INSTANCE with matching:\nT5 -> T59\nT7 -> T61", fontsize=14, style = filled, fillcolor = lightgrey];
61 -> 118 [arrowhead = none , style=dashed];
118 -> 5[style=dashed];

119 [label="EVAL with clause\npdl(nil, X92) :- ','(!_7, eq(X92, -(X93, X93))).\nand substitutionT57 -> nil,\nT37 -> T66,\nX37 -> X94,\nX92 -> -(T66, X94),\nT65 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 119 [arrowhead = none ];
119 -> 70;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 120 [arrowhead = none ];
120 -> 71;

121 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 121 [arrowhead = none ];
121 -> 72;

122 [label="EVAL with clause\npdl(X114, -(.(X115, X116), X117)) :- ','(value(X114, X115), ','(left(X114, X118), ','(right(X114, X119), ','(pdl(X118, -(X116, X120)), pdl(X119, -(X120, X117)))))).\nand substitutionT57 -> T76,\nX114 -> T76,\nX115 -> T79,\nX116 -> T80,\nT37 -> .(T79, T80),\nX37 -> X121,\nX117 -> X121,\nT77 -> T79,\nT78 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 122 [arrowhead = none ];
122 -> 79;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 123 [arrowhead = none ];
123 -> 80;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 124 [arrowhead = none ];
124 -> 73;

125 [label="ONLY EVAL with clause\neq(X99, X99).\nand substitutionT66 -> T69,\nX94 -> T69,\nX99 -> -(T69, T69),\nX93 -> T69,\nX100 -> T69", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 125 [arrowhead = none ];
125 -> 75;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 126 [arrowhead = none ];
126 -> 76;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 127 [arrowhead = none ];
127 -> 81;

128 [label="BACKTRACK\nfor clause: value(nil, X5)\nwith clash: (pdl(T76, -(T37, X37)), pdl(nil, X92))", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 128 [arrowhead = none ];
128 -> 82;

129 [label="EVAL with clause\nvalue(tree(X132, X133, X134), X133).\nand substitutionX132 -> T89,\nX133 -> T90,\nX134 -> T91,\nT76 -> tree(T89, T90, T91),\nT79 -> T90,\nT80 -> T92", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 129 [arrowhead = none ];
129 -> 83;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 130 [arrowhead = none ];
130 -> 84;

131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
83 -> 131 [arrowhead = none ];
131 -> 85;

132 [label="BACKTRACK\nfor clause: left(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
85 -> 132 [arrowhead = none ];
132 -> 86;

133 [label="ONLY EVAL with clause\nleft(tree(X147, X148, X149), X147).\nand substitutionT89 -> T101,\nX147 -> T101,\nT90 -> T102,\nX148 -> T102,\nT91 -> T103,\nX149 -> T103,\nX118 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 133 [arrowhead = none ];
133 -> 87;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
87 -> 134 [arrowhead = none ];
134 -> 88;

135 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 135 [arrowhead = none ];
135 -> 89;

136 [label="ONLY EVAL with clause\nright(tree(X162, X163, X164), X164).\nand substitutionT101 -> T112,\nX162 -> T112,\nT102 -> T113,\nX163 -> T113,\nT103 -> T114,\nX164 -> T114,\nX119 -> T114", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 136 [arrowhead = none ];
136 -> 90;

137 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
90 -> 137 [arrowhead = none ];
137 -> 91;

138 [label="SPLIT 2\nnew knowledge:\nT112 is ground\nreplacements:X120 -> T116", fontsize=14, style = filled, fillcolor = violet];
90 -> 138 [arrowhead = none ];
138 -> 92;

139 [label="INSTANCE with matching:\nT57 -> T112\nT37 -> T92\nX37 -> X120", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 139 [arrowhead = none , style=dashed];
139 -> 60[style=dashed];

140 [label="INSTANCE with matching:\nT57 -> T114\nT37 -> T116\nX37 -> X121", fontsize=14, style = filled, fillcolor = lightgrey];
92 -> 140 [arrowhead = none , style=dashed];
140 -> 60[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f2_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f57_in(z0, z2), tree(z0, z1, z2))
U2(f57_out1, tree(z0, z1, z2)) → f5_out1
f60_in(nil) → f60_out1
f60_in(tree(z0, z1, z2)) → U3(f90_in(z0, z2), tree(z0, z1, z2))
U3(f90_out1, tree(z0, z1, z2)) → f60_out1
f57_in(z0, z1) → U4(f60_in(z0), z0, z1)
U4(f60_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f57_out1
f90_in(z0, z1) → U6(f60_in(z0), z0, z1)
U6(f60_out1, z0, z1) → U7(f60_in(z1), z0, z1)
U7(f60_out1, z0, z1) → f90_out1

Tuples:

F2_IN(z0) → c(U1'(f5_in(z0), z0), F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(U2'(f57_in(z0, z2), tree(z0, z1, z2)), F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(U3'(f90_in(z0, z2), tree(z0, z1, z2)), F90_IN(z0, z2))
F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
U4'(f60_out1, z0, z1) → c9(U5'(f5_in(z1), z0, z1), F5_IN(z1))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
U6'(f60_out1, z0, z1) → c12(U7'(f60_in(z1), z0, z1), F60_IN(z1))

S tuples:

F2_IN(z0) → c(U1'(f5_in(z0), z0), F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(U2'(f57_in(z0, z2), tree(z0, z1, z2)), F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(U3'(f90_in(z0, z2), tree(z0, z1, z2)), F90_IN(z0, z2))
F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
U4'(f60_out1, z0, z1) → c9(U5'(f5_in(z1), z0, z1), F5_IN(z1))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
U6'(f60_out1, z0, z1) → c12(U7'(f60_in(z1), z0, z1), F60_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f5_in, U2, f60_in, U3, f57_in, U4, U5, f90_in, U6, U7

Defined Pair Symbols:

F2_IN, F5_IN, F60_IN, F57_IN, U4', F90_IN, U6'

Compound Symbols:

c, c3, c6, c8, c9, c11, c12

(17) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f2_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f57_in(z0, z2), tree(z0, z1, z2))
U2(f57_out1, tree(z0, z1, z2)) → f5_out1
f60_in(nil) → f60_out1
f60_in(tree(z0, z1, z2)) → U3(f90_in(z0, z2), tree(z0, z1, z2))
U3(f90_out1, tree(z0, z1, z2)) → f60_out1
f57_in(z0, z1) → U4(f60_in(z0), z0, z1)
U4(f60_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f57_out1
f90_in(z0, z1) → U6(f60_in(z0), z0, z1)
U6(f60_out1, z0, z1) → U7(f60_in(z1), z0, z1)
U7(f60_out1, z0, z1) → f90_out1

Tuples:

F5_IN(tree(z0, z1, z2)) → c3(U2'(f57_in(z0, z2), tree(z0, z1, z2)), F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(U3'(f90_in(z0, z2), tree(z0, z1, z2)), F90_IN(z0, z2))
F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
U4'(f60_out1, z0, z1) → c9(U5'(f5_in(z1), z0, z1), F5_IN(z1))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
U6'(f60_out1, z0, z1) → c12(U7'(f60_in(z1), z0, z1), F60_IN(z1))
F2_IN(z0) → c1(U1'(f5_in(z0), z0))
F2_IN(z0) → c1(F5_IN(z0))

S tuples:

F5_IN(tree(z0, z1, z2)) → c3(U2'(f57_in(z0, z2), tree(z0, z1, z2)), F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(U3'(f90_in(z0, z2), tree(z0, z1, z2)), F90_IN(z0, z2))
F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
U4'(f60_out1, z0, z1) → c9(U5'(f5_in(z1), z0, z1), F5_IN(z1))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
U6'(f60_out1, z0, z1) → c12(U7'(f60_in(z1), z0, z1), F60_IN(z1))
F2_IN(z0) → c1(U1'(f5_in(z0), z0))
F2_IN(z0) → c1(F5_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f5_in, U2, f60_in, U3, f57_in, U4, U5, f90_in, U6, U7

Defined Pair Symbols:

F5_IN, F60_IN, F57_IN, U4', F90_IN, U6', F2_IN

Compound Symbols:

c3, c6, c8, c9, c11, c12, c1

(19) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f2_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f57_in(z0, z2), tree(z0, z1, z2))
U2(f57_out1, tree(z0, z1, z2)) → f5_out1
f60_in(nil) → f60_out1
f60_in(tree(z0, z1, z2)) → U3(f90_in(z0, z2), tree(z0, z1, z2))
U3(f90_out1, tree(z0, z1, z2)) → f60_out1
f57_in(z0, z1) → U4(f60_in(z0), z0, z1)
U4(f60_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f57_out1
f90_in(z0, z1) → U6(f60_in(z0), z0, z1)
U6(f60_out1, z0, z1) → U7(f60_in(z1), z0, z1)
U7(f60_out1, z0, z1) → f90_out1

Tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F2_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F2_IN(z0) → c1

S tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F2_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F2_IN(z0) → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f5_in, U2, f60_in, U3, f57_in, U4, U5, f90_in, U6, U7

Defined Pair Symbols:

F57_IN, F90_IN, F2_IN, F5_IN, F60_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(21) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f2_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f57_in(z0, z2), tree(z0, z1, z2))
U2(f57_out1, tree(z0, z1, z2)) → f5_out1
f60_in(nil) → f60_out1
f60_in(tree(z0, z1, z2)) → U3(f90_in(z0, z2), tree(z0, z1, z2))
U3(f90_out1, tree(z0, z1, z2)) → f60_out1
f57_in(z0, z1) → U4(f60_in(z0), z0, z1)
U4(f60_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f57_out1
f90_in(z0, z1) → U6(f60_in(z0), z0, z1)
U6(f60_out1, z0, z1) → U7(f60_in(z1), z0, z1)
U7(f60_out1, z0, z1) → f90_out1

Tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F2_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F2_IN(z0) → c1

S tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))

K tuples:

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1

Defined Rule Symbols:

f2_in, U1, f5_in, U2, f60_in, U3, f57_in, U4, U5, f90_in, U6, U7

Defined Pair Symbols:

F57_IN, F90_IN, F2_IN, F5_IN, F60_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))

We considered the (Usable) Rules:

f60_in(nil) → f60_out1
f60_in(tree(z0, z1, z2)) → U3(f90_in(z0, z2), tree(z0, z1, z2))
f90_in(z0, z1) → U6(f60_in(z0), z0, z1)
U3(f90_out1, tree(z0, z1, z2)) → f60_out1
U6(f60_out1, z0, z1) → U7(f60_in(z1), z0, z1)
U7(f60_out1, z0, z1) → f90_out1

And the Tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F2_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F2_IN(z0) → c1

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F2_IN(x₁)) = [1] + [2]x₁   
POL(F57_IN(x₁, x₂)) = [2]x₂   
POL(F5_IN(x₁)) = [2]x₁   
POL(F60_IN(x₁)) = 0   
POL(F90_IN(x₁, x₂)) = 0   
POL(U3(x₁, x₂)) = 0   
POL(U4'(x₁, x₂, x₃)) = [2]x₃   
POL(U6(x₁, x₂, x₃)) = [2]x₃   
POL(U6'(x₁, x₂, x₃)) = 0   
POL(U7(x₁, x₂, x₃)) = x₃   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f60_in(x₁)) = 0   
POL(f60_out1) = 0   
POL(f90_in(x₁, x₂)) = [2] + [2]x₂   
POL(f90_out1) = 0   
POL(nil) = 0   
POL(tree(x₁, x₂, x₃)) = [2] + x₃   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f2_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f57_in(z0, z2), tree(z0, z1, z2))
U2(f57_out1, tree(z0, z1, z2)) → f5_out1
f60_in(nil) → f60_out1
f60_in(tree(z0, z1, z2)) → U3(f90_in(z0, z2), tree(z0, z1, z2))
U3(f90_out1, tree(z0, z1, z2)) → f60_out1
f57_in(z0, z1) → U4(f60_in(z0), z0, z1)
U4(f60_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f57_out1
f90_in(z0, z1) → U6(f60_in(z0), z0, z1)
U6(f60_out1, z0, z1) → U7(f60_in(z1), z0, z1)
U7(f60_out1, z0, z1) → f90_out1

Tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F2_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F2_IN(z0) → c1

S tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))

K tuples:

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))

Defined Rule Symbols:

f2_in, U1, f5_in, U2, f60_in, U3, f57_in, U4, U5, f90_in, U6, U7

Defined Pair Symbols:

F57_IN, F90_IN, F2_IN, F5_IN, F60_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(25) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f2_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f57_in(z0, z2), tree(z0, z1, z2))
U2(f57_out1, tree(z0, z1, z2)) → f5_out1
f60_in(nil) → f60_out1
f60_in(tree(z0, z1, z2)) → U3(f90_in(z0, z2), tree(z0, z1, z2))
U3(f90_out1, tree(z0, z1, z2)) → f60_out1
f57_in(z0, z1) → U4(f60_in(z0), z0, z1)
U4(f60_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f57_out1
f90_in(z0, z1) → U6(f60_in(z0), z0, z1)
U6(f60_out1, z0, z1) → U7(f60_in(z1), z0, z1)
U7(f60_out1, z0, z1) → f90_out1

Tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F2_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F2_IN(z0) → c1

S tuples:

F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))

K tuples:

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))

Defined Rule Symbols:

f2_in, U1, f5_in, U2, f60_in, U3, f57_in, U4, U5, f90_in, U6, U7

Defined Pair Symbols:

F57_IN, F90_IN, F2_IN, F5_IN, F60_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))

We considered the (Usable) Rules:

f60_in(nil) → f60_out1
f60_in(tree(z0, z1, z2)) → U3(f90_in(z0, z2), tree(z0, z1, z2))
f90_in(z0, z1) → U6(f60_in(z0), z0, z1)
U3(f90_out1, tree(z0, z1, z2)) → f60_out1
U6(f60_out1, z0, z1) → U7(f60_in(z1), z0, z1)
U7(f60_out1, z0, z1) → f90_out1

And the Tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F2_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F2_IN(z0) → c1

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F2_IN(x₁)) = [3]x₁   
POL(F57_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(F5_IN(x₁)) = [3]x₁   
POL(F60_IN(x₁)) = x₁   
POL(F90_IN(x₁, x₂)) = x₁ + x₂   
POL(U3(x₁, x₂)) = 0   
POL(U4'(x₁, x₂, x₃)) = [3]x₃   
POL(U6(x₁, x₂, x₃)) = [2] + [2]x₂   
POL(U6'(x₁, x₂, x₃)) = x₃   
POL(U7(x₁, x₂, x₃)) = [2] + x₂   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f60_in(x₁)) = 0   
POL(f60_out1) = 0   
POL(f90_in(x₁, x₂)) = [2] + [2]x₁   
POL(f90_out1) = [2]   
POL(nil) = 0   
POL(tree(x₁, x₂, x₃)) = [1] + x₁ + x₃   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f5_in(z0), z0)
U1(f5_out1, z0) → f2_out1
f5_in(nil) → f5_out1
f5_in(tree(z0, z1, z2)) → U2(f57_in(z0, z2), tree(z0, z1, z2))
U2(f57_out1, tree(z0, z1, z2)) → f5_out1
f60_in(nil) → f60_out1
f60_in(tree(z0, z1, z2)) → U3(f90_in(z0, z2), tree(z0, z1, z2))
U3(f90_out1, tree(z0, z1, z2)) → f60_out1
f57_in(z0, z1) → U4(f60_in(z0), z0, z1)
U4(f60_out1, z0, z1) → U5(f5_in(z1), z0, z1)
U5(f5_out1, z0, z1) → f57_out1
f90_in(z0, z1) → U6(f60_in(z0), z0, z1)
U6(f60_out1, z0, z1) → U7(f60_in(z1), z0, z1)
U7(f60_out1, z0, z1) → f90_out1

Tuples:

F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
F2_IN(z0) → c1(F5_IN(z0))
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F2_IN(z0) → c1

S tuples:

F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))

K tuples:

F2_IN(z0) → c1(F5_IN(z0))
F2_IN(z0) → c1
F5_IN(tree(z0, z1, z2)) → c3(F57_IN(z0, z2))
F57_IN(z0, z1) → c8(U4'(f60_in(z0), z0, z1), F60_IN(z0))
U4'(f60_out1, z0, z1) → c9(F5_IN(z1))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))

Defined Rule Symbols:

f2_in, U1, f5_in, U2, f60_in, U3, f57_in, U4, U5, f90_in, U6, U7

Defined Pair Symbols:

F57_IN, F90_IN, F2_IN, F5_IN, F60_IN, U4', U6'

Compound Symbols:

c8, c11, c1, c3, c6, c9, c12, c1

(29) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F90_IN(z0, z1) → c11(U6'(f60_in(z0), z0, z1), F60_IN(z0))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))
U6'(f60_out1, z0, z1) → c12(F60_IN(z1))
F60_IN(tree(z0, z1, z2)) → c6(F90_IN(z0, z2))

Now S is empty

(30) BOUNDS(O(1), O(1))